Algebraic convergence of function groups

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the rectangles are called rails, the vertical segments ties and the intersections of two rectangles switches. A lamination is carried by τ if it is contained in τ and.
Comment. Math. Helv. 77 (2002) 244–269 0010-2571/02/020244-26 $ 1.50+0.20/0

c 2002 Birkh¨ ° auser Verlag, Basel

Commentarii Mathematici Helvetici

Algebraic convergence of function groups Gero Kleineidam and Juan Souto

Abstract. We give a sufficient condition for a sequence of convex cocompact hyperbolic structures on a fixed compression body to have an algebraically convergent subsequence. This extends a result of Otal. Further if the manifold is a handlebody we show that certain laminations play a similar role in deformation space as binding curves in Teichm¨ uller theory. Mathematics Subject Classification (2000). 30F40, 20E08, 57M50. Keywords. Hyperbolic manifolds, compression bodies, algebraic convergence, Masur domain, R-trees.

1. Introduction A compression body N is a compact 3-manifold which is the connected sum along the boundary on a closed ball of solid tori and trivial interval bundles over closed surfaces of genus at least 2. Throughout the paper, we only consider the case that the fundamental group π1 (N ) splits as a non-trivial free product. Equivalently, we rule out that N is a trivial interval bundle over a closed surface or a solid torus. In particular, the boundary ∂N has a unique compressible component which is called the exterior boundary ∂e N . For more on the topology of compression bodies see Bonahon [Bon83, Appendix B]. Using Klein-combination one can construct a convex cocompact representation ρ0 of π1 (N ) into PSL2 C such that H3 /ρ0 (π1 (N )) is homeomorphic to the interior of the compression body N [MT98]. Such a representation is said to uniformize N . The exterior boundary ∂e N is covered by a connected component of the disˆ which is invariant under ρ0 (π1 (N )). Kleinian groups continuity domain Ωρ0 ⊂ C, having an invariant component of the discontinuity domain are called function groups. The quotient of H3 by any convex cocompact function group is homeomorphic to the interior of a compression body. See Maskit [Mas88] for more on function groups. Due to a theorem of Marden [And98, MT98], every quasi-conformal deformation of ρ0 uniformizes N , too. By Ahlfors–Bers theory, QH(ρ0 ), the space of The authors were partially supported by the Sonderforschungsbereich 256.

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quasi-conformal deformations of ρ0 up to conjugation by an element of PSL2 C, is parameterized by the Teichm¨ uller space T (∂N ). More precisely, there is a normal covering, called the Ahlfors–Bers map T (∂N ) → QH(ρ0 ) whose deck transformation group is the group of isotopy classes of diffeomorphisms of N which are homotopic to the identity (see [MT98]). Note that the Teichm¨ uller space T (∂N ) can be identified with T (∂e N ) × T (∂1 N ) × · · · × T (∂k N ) where ∂e N, ∂1 N, . . . , ∂k N are the boundary components of N. As homeomorphisms of N preserve ∂e N, the deck transformation group of the Ahlfors–Bers map acts on T (∂e N ). The space QH(ρ0 ) is contained in the deformation space of π1 (N ), the space of PSL2 C-conjugacy classes of discrete and faithful representations of π1 (N ) into PSL2 C. The compact-open topology on it is the so-called algebraic topology. With respect to this topology, QH(ρ0 ) is open in deformation space. For more on the deformation theory of Kleinian groups, see [And98, MT98]. Since deformation space is not compact, it is an interesting question to determine when a divergent sequence in QH(ρ0 ) converges in deformation space. Canary [Can91] showed that for all compact subsets K ⊂ T (∂e N ) the image under the Ahlfors–Bers map of K ×T (∂1 N )×· · ·×T (∂k N ) has compact closure in deformation space. The goal of this paper is to study sequences of quasi-conformal deformations of ρ0 such that the corresponding sequences in T (∂e N ) diverge. Thurston [FLP79] compactified Teichm¨ uller space via PML, the space of projective classes of measured laminations. Masur [Mas86] and Otal [Ota88] studied the dynamics of the mapping class group of the compression body N on the space of projective classes of measured laminations on the exterior boundary ∂e N and described an open set O ⊂ PML on which the action is properly discontinuous. The set O is called the Masur domain. Otal [Ota88] analyzed further geometric properties of laminations in O. In particular, he proved that given a convex cocompact representation uniformizing N , every lamination λ in the Masur domain is realized by a pleated surface. We will say that a sequence (ρi ) in QH(ρ0 ) converges into the Masur domain if it is parameterized under the Ahlfors–Bers map by a sequence (Sie , Si1 , . . . , Sik )i ⊂ T (∂N ) such that (Sie )i converges to a measured lamination λ ∈ O. We impose no restrictions on the conformal structures on the incompressible boundary components. By abuse, we will say that the sequence (ρi ) ⊂ QH(ρ0 ) converges to λ ∈ O. Conjecture (Thurston). Let ρ0 be a convex cocompact representation uniformizing a compression body N . If (ρi )i is a sequence in QH(ρ0 ) converging to a lamination in the Masur domain, then it has a convergent subsequence in deformation space. Canary [Can93] proved the conjecture under the extra assumption that N is a handlebody and that there is an identification of N with the trivial interval bundle

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over a surface Σ with boundary ∂Σ such that the lengths of the geodesics in the free homotopy classes of ∂Σ remain bounded with respect to the representations ρi . In case N is the connected sum along the boundary of two trivial bundles over closed surfaces, Ohshika [Ohs97] gave a partial answer to the above conjecture. Otal [Ota94] proved that the conjecture holds for handlebodies of genus two and laminations with simply connected complementary regions. Such laminations are also called minimal arational. We follow the strategy of Otal’s proof and show for general compression bodies. Theorem 1. Let ρ0 be a convex cocompact representation uniformizing a compression body N . If (ρi )i is a sequence in QH(ρ0 ) converging to a minimal arational lamination in the Masur domain, then it has a convergent subsequence in deformation space. Further, we prove that laminations in the Masur domain play a similar role for handlebodies as binding curves do in Teichm¨ uller theory. This generalizes Theorem 6.1 in Canary [Can93]. Theorem 2. Let N be a handlebody and λ a measured lamination in the Masur domain. The set of convex cocompact representations ρ uniformizing N such that lρ (λ), the length of λ with respect to ρ, is less than a constant C > 0 is precompact in deformation space. The proofs of Theorem 1 and Theorem 2 follow the same lines. We restrict ourselves to a brief outline of the proof of Theorem 1. Seeking for a contradiction, let (ρi ) be a sequence in QH(ρ0 ) which converges to a minimal arational lamination λ ∈ O but does not contain any convergent subsequence in deformation space. By Theorems of Thurston [FLP79] and Canary [Can91], convergence to λ implies that there is a sequence of curves (γi ) on ∂e N converging to λ in PML such that the ratios of the translation lengths in H3 of ρi (γi ) and ρ0 (γi ) tend to 0. By a Theorem of Morgan and Shalen [MS84], divergence in deformation space implies that a subsequence of (ρi ), say the whole sequence, converges in an appropriate sense to a minimal and small action of π1 (N ) on an R-tree T . The lamination ˜ the λ is said to be realized in T if there is a continuous equivariant map from λ, 2 ˜ lift of λ to H , to T which is injective on each leaf of λ. Otal [Ota94] proved that if λ is realized in T , then for every sequence of curves (γi ) converging to λ in PML, the ratios of the translation lengths in H3 of ρi (γi ) and ρ0 (γi ) tend to ∞. Hence, we obtain the desired contradiction by proving that every minimal arational lamination in the Masur domain is realized in every R-tree which admits a minimal small action of π1 (N ). This was previously established by Otal [Ota94] in the case of the handlebody of genus 2. He made use of a Theorem of Culler and Vogtmann [CV91] which gives a geometric description of all small actions on R-trees of the free group of rank 2. It is known that such a characterization is not

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possible in general, not even for free groups of higher rank [GL95]. Our approach is different. Suppose that a minimal arational lamination λ ∈ O and an R-tree T that admits a small action of π1 (N ) are given. We show that there is a measured lamination µ on the exterior boundary ∂e N and a morphism from the dual tree Tµ to T such that the composition of the projection H2 → Tµ with ˜ Due to the morphism Tµ → T is monotone and non-constant on every leaf of λ. a result of Otal [Ota94], such a map can be homotoped to a realization of λ in T . The existence of µ ensues from a detailed study of morphisms from dual trees to T . This part is motivated by ideas of Skora [Sko96]. The paper is structured as follows: In section 2, we review some facts about trees, laminations, and divergence of representations. In Section 3, we reduce Theorem 1 and Theorem 2 to statements on realizations of laminations in R-trees. Section 4 is devoted to the analysis of laminations on the exterior boundary of N . The techniques are motivated by earlier work of Otal [Ota88]. In section 5, we study morphisms from dual trees to R-trees which admit a minimal small action of π1 (N ). In section 6, we construct the desired realizations using the results of section 4 and section 5. The authors would like to express their special gratitude to Professor Ursula Hamenst¨ adt and Professor Jean-Pierre Otal for their patience, their encouragement and the fruitful discussions with them. There is no doubt that the present paper would not have been possible without Otal’s fundamental work on this topic. The first author wants to thank Professor Fr´ed´eric Paulin for his invitation to a two-month-stay in Orsay. We thank the referee for a careful reading and useful suggestions.

2. Preliminaries In this section, we review some facts about trees, laminations and divergence of representations. Let N be a compression body. After the choice of a basepoint ? on the exterior boundary ∂e N , we have a surjective homomorphism ϕ : π1 (∂e N, ?) → π1 (N, ?) which we call compression homomorphism. If there is no risk of confusion, we will use the symbol γ for elements in π1 (∂e N, ?) as well as for their images under ϕ. Further for simplicity, we often write π1 (∂e N ) and π1 (N ). Later it will be useful to view every action π1 (N ) y X of π1 (N ) on a space X as a π1 (∂e N )-action, too: π1 (∂e N ) × X → X,

(g, x) 7→ ϕ(g)x

All actions on metric spaces (X, dX ) will be isometric. The translation length of an isometry g of a metric space X is defined to be inf{dX (x, gx)|x ∈ X}. The metric spaces we are going to work with are H2 , H3 and R-trees.

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2.1. R-trees An R-tree T is a path metric space such that any two points p, q can be joined by a unique arc. There is a classification of the isometries of an R-tree T . An isometry g : T → T has either translation length lT (g) = 0 and has a fixed point or lT (g) > 0 and there is a unique invariant geodesic line in T , the axis of g. An action G y T is called minimal if there is no proper invariant subtree. An action G y T is called small if the stabilizer of every non-degenerate arc is virtually abelian. More about R-trees can be found in Kapovich [Kap00]. Morgan and Shalen [MS84] used R-trees to compactify the deformation space. They use algebraic methods, for a more geometric approach see Bestvina [Bes88] and Paulin [Pau88]. Compactness Theorem (Morgan–Shalen). Let G be a finitely generated group containing a free group of rank 2 and let ρi : G → PSL2 C be a sequence of discrete and faithful representations. Then after passing to a subsequence either (1) (ρi ) converges in the deformation space of G, or (2) there is a minimal small action G y T on an R-tree and a sequence of real numbers ²i → 0 with lim ²i lρi (γ) = lT (γ) i

for all γ ∈ G, where lρi (γ) is the translation length of ρi (γ) in H3 . We remark that a minimal small action of such a group G on a tree is characterized by the translation lengths of the elements of G [Kap00]. This allows us to say that the sequence (ρi )i converges to the action G y T . We will apply the Compactness Theorem to the case that G is the fundamental group of a compression body. 2.2. Laminations A lamination on a closed hyperbolic surface S is a compact subset of S which can be decomposed as a disjoint union of simple geodesics, called leaves. The sets of laminations with respect to two hyperbolic structures on the same surface can be naturally identified (see [CB88]); so a lamination can be considered as a topological object. A lamination is called minimal if every half-leaf is dense. Each lamination can be decomposed as a union of finitely many connected minimal laminations, called minimal components, and finitely many non-compact isolated leaves. The set of laminations is compact with respect to the topology induced by the Hausdorff distance. We will refer to this topology as the Hausdorff topology.

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A train-track τ in S is a finite union of rectangles with “long” horizontal sides which are foliated by “short” vertical geodesic segments and which meet only at non-degenerate segments contained in the vertical sides. The horizontal sides of the rectangles are called rails, the vertical segments ties and the intersections of two rectangles switches. A lamination is carried by τ if it is contained in τ and transverse to the ties. If λ is carried by τ , then the set of laminations carried by τ forms a neighbourhood of λ with respect to the Hausdorff topology [Ota96]. A measured lamination is a lamination with a transverse measure of full support. The support of a measured lamination is a finite union of minimal components, in particular it does not contain any isolated non-compact leaf. A minimal lamination is called minimal arational if its complementary regions are simplyconnected. There is a topology on the set ML of measured laminations which is induced by the intersection form i : ML × ML → R+ [Ota96, FLP79]. Rescaling the measure provides an action of R+ on ML. The quotient with the quotient topology is the space of projective measured laminations and is denoted PML. It is compact. If a sequence of projective measured laminations converges to a projective measured lamination λ in PML and to a lamination λH in the Hausdorff topology, then λ – or more precisely the support of λ – is contained in λH . The Teichm¨ uller space of a closed surface S is denoted by T (S). Thurston [Thu86] studied the length function on the space of measured laminations on S. It is the unique continuous function T (S) × ML −→ R+ ,

(σ, λ) 7→ lσ (λ)

which extends the function that associates to a point σ ∈ T (S) and to a weighted simple closed geodesic a · γ, a > 0, the length of γ in σ multiplied by a. The Teichm¨ uller space T (S) can be compactified by the space PML of projective measured laminations on S [FLP79]. This compactification reflects the geometric behaviour of divergent sequences in T (S). In particular, if a sequence (Si ) ⊂ T (S) converges to λ ∈ PML, then there is a sequence (γi ) of simple closed curves converging to λ in PML and such that lSi (γi )/lS0 (γi ) −→ 0 for i −→ ∞. However this does not imply that the lengths lSi (λ) tend to 0. Indeed, take an element of T (S) and iterate a Dehn twist about a fixed curve γ on it. The resulting sequence tends to γ, seen as an element of PML, but the length of γ is constant during the sequence. Given a hyperbolic structure on the compression body N , a pleated surface is a length preserving map f : S → N from a hyperbolic surface S ∈ T (∂e N ) to N homotopic to the inclusion ∂e N ,→ N and such that each point p ∈ S is contained in a geodesic segment which is mapped isometrically. A lamination λ on the exterior boundary ∂e N is realized by a pleated surface if there is a pleated surface that maps each leaf of λ to a geodesic in N . Notice that a realization of a lamination λ by a pleated surface induces a map from λ to the projectivized

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tangent bundle of N . If ρ is a representation of π1 (N ) into PSL2 C uniformizing N and λ is a measured lamination on the exterior boundary that is realized by a pleated surface, we can define lρ (λ), the length of λ with respect to ρ, to be its length with respect to the hyperbolic structure of the pleated surface realizing λ. See Otal [Ota96, Appendix] for more about laminations and measured laminations and Fathi–Laudenbach–Po´enaru [FLP79] for a detailed exposition of Thurston’s compactification of Teichm¨ uller space. Pleated surfaces are discussed in [Thu86, Ota88, Kap00].

2.3. Dual trees For a measured lamination µ on a surface S, we denote by µ ˜ its lift to H2 . If µ 2 does not have atoms, the semidistance on H induced by integrating the transverse measure µ ˜ along paths is continuous with respect to the usual topology of H2 . The support of an atom is a closed geodesic, therefore it is possible to avoid atoms by replacing closed leaves in µ by annuli foliated by parallel closed curves. Denote by Fµ the measured partial foliation that we obtain by this process and by Feµ the lift of Fµ to H2 . The quotient of H2 under the semi-distance induced by Fµ depends only on µ, it is denoted Tµ , and the projection πFµ : H2 → Tµ is continuous. Tµ is an R-tree, called the dual tree of µ. The fundamental group of the surface S acts on Tµ and the action is small and minimal. Dual trees are discussed in detail by Otal [Ota96] and Kapovich [Kap00]. Using dual trees, Skora [Sko96] established a 1-1-correspondence between minimal small actions of the fundamental group of a closed surface S and measured laminations on S. Theorem (Skora). Let π1 (S) y T be a minimal and small action of π1 (S) on an R-tree T , then there is a unique µ ∈ ML and an equivariant isometry Tµ → T . Skora’s ideas will be used in section 5 where we study certain maps from trees dual to laminations on the exterior boundary of a compression body N to a given R-tree with a minimal small action of π1 (N ).

3. Main Theorems The goal of this section is to reduce Theorem 1 and Theorem 2 to a property of minimal small actions of the fundamental group of a compression body N on R-trees. A simple closed curve m on the exterior boundary ∂e N which is homotopically trivial in N but not in ∂e N is called a meridian. Note that by Dehn’s Lemma

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[Jac80] every meridian bounds an embedded disk in N . Recall that we use the symbol γ for elements in π1 (∂e N ) as well as for their images under the compression homomorphism ϕ : π1 (∂e N ) → π1 (N ). A meridian may be seen as an element of PML, too. The set of projective classes of weighted multicurves of meridians in PML will be denoted by M and its closure in PML by M 0 (see Otal [Ota88]). N is a small compression body if it is the connected sum along the boundary of either two trivial interval bundles over closed surfaces or an interval bundle over a closed surface and a solid torus. For a small compression body, set O := {λ ∈ PML| i(λ, µ) > 0 for all µ ∈ PML such that there is ν ∈ M 0 with i(µ, ν) = 0} If N is not a small compression body, set O := {λ ∈ PML| i(λ, µ) > 0 for all µ ∈ M 0 } The set O is called the Masur domain and is open by continuity of the intersection form and compactness of PML. We will say that λ ∈ ML is in O (resp. M 0 ) if its projective class is in O (resp. M 0 ). Otal [Ota88] proved (see also Ohshika [Ohs]) Theorem on pleated surfaces (Otal). Let N be a compression body with a convex cocompact hyperbolic structure. Every lamination λ ⊂ ∂e N containing the support of a measured lamination in the Masur domain is realized by a pleated surface in N . Moreover, the induced map from λ to the projectivized tangent bundle of N is a homeomorphism onto its image Pλ . The image in Pλ of a leaf of λ is the trace of a geodesic and is equally called a eλ the preimage of Pλ in the projectivized tangent bundle of H3 . leaf. Denote by P The following definition is due to Otal [Ota94]: Definition. Let N be a compression body with a convex cocompact hyperbolic structure and π1 (N ) y T an action on an R-tree T . A lamination λ ⊂ ∂e N is realized in T if there is a continuous π1 (N )-equivariant map eλ → T Φ: P which is injective when restricted to any leaf. eλ , this definition is equivalent to the As λ is mapped homeomorphically onto P definition of realization given in the introduction. The following Theorem was proved by Otal [Ota94] in the case that N is a handlebody, but a careful checking of the proof shows that it holds for compression bodies as well.

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Continuity Theorem (Otal). Let (ρi ) be a sequence in the deformation space of π1 (N ) converging to a non-trivial minimal small action π1 (N ) y T and (γi ) a sequence of simple closed curves on ∂e N converging in PML to a minimal arational measured lamination λ in O which is realized in T . Then for all N > 0 there is iN > 0 such that lρi (γi ) ≥ N lρ0 (γi ) for all i ≥ iN where lρ (γ) denotes the translation length of ρ(ϕ(γ)) in H3 . Remark. It seems to be a delicate issue to extend the theorem to all sequences of curves (γi ) converging in PML to an arbitrary lamination λ in O which is realized in the tree T . On the other hand, the conclusion of the theorem is valid for an arbitrary sequence of curves (γi ) provided every Hausdorff limit of (γi ) is realized in T . Indeed, the latter condition is the only one used in the proof of the theorem, and it is weaker than the one stated in the theorem (see [Ota94]). In the last section we will prove Theorem 3. Let π1 (N ) y T be a non-trivial minimal small action on an R-tree T and λ a minimal arational measured lamination in the Masur domain, then λ is realized in T . Next we reduce Theorem 1 to Theorem 3 (see Otal [Ota94]). Theorem 1. Let ρ0 be a convex cocompact representation uniformizing a compression body N . If (ρi )i is a sequence in QH(ρ0 ) converging to a minimal arational lamination in the Masur domain, then it has a convergent subsequence in deformation space. Proof. Seeking a contradiction, suppose that (ρi )i converges to a minimal and small action π1 (N ) y T on an R-tree. By definition, there is a sequence (Sie , Si1 , . . . , Sik ) ⊂ T (∂N ) which is mapped to (ρi ) under the Ahlfors–Bers map and such that the sequence (Sie ) ⊂ T (∂e N ) converges to λ. Then there is a sequence (γi ) of simple closed curves on the exterior boundary ∂e N converging to λ in PML with lSie (γi )/lS0e (γi ) −→ 0 for i −→ ∞.

(1)

On the other hand, for all A > 0 there is some iA such that lSie (m) > A for all i ≥ iA and all meridians m. Then by a Theorem due to Canary [Can91] there is K > 0 such that for all i ≥ iA lρi (γi ) ≤ K lSie (γi )

(2)

where lρi (γi ) is the translation length of ρi (ϕ(γi )) in H3 . Combining equation (1) and equation (2) we deduce with the same arguments as Canary [Can93] in the

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handlebody case that lρi (γi )/lρ0 (γi ) −→ 0 for i −→ ∞. By the Continuity Theorem, λ is not realized in T , contradicting Theorem 3.

¤

In the case that N is a handlebody we will show in section 6 Corollary 3. Let N be a handlebody and π1 (N ) y T be a non-trivial minimal small action on an R-tree T . At least one minimal component of every measured lamination in the Masur domain is realized in T . We reduce Theorem 2 to Corollary 3. Theorem 2. Let N be a handlebody and λ a measured lamination in the Masur domain. The set of convex cocompact representations ρ uniformizing N such that lρ (λ), the length of λ with respect to ρ, is less than a constant C > 0 is precompact in deformation space. Proof. Suppose again that there is a sequence (ρi ) of convex cocompact representations uniformizing N with lρi (λ) < C that converges to some non-trivial minimal small action π1 (N ) y T on an R-tree. The length of any minimal component λ0 of λ is also bounded by C for all i. By Otal’s Theorem on pleated surfaces, the lamination λ is realized by a pleated surface in H3 /ρi (π1 (N )) for all i. Hence in each neighbourhood of λ0 with respect to the Hausdorff topology we find a simple closed curve γi with lρi (γi ) < C lρ0 (γi ). By a diagonal argument, we can assume that the sequence (γi ) converges to λ0 in the Hausdorff topology. The Continuity Theorem applies to this sequence by the remark after it; thus, the lamination λ0 cannot be realized in T . As λ0 was arbitrary this contradicts Corollary 3. ¤

4. Laminations on the exterior boundary Let ρ0 : π1 (N ) → PSL2 C be a convex cocompact representation uniformizing the compression body N . The image of ρ0 is a function group. Following Otal’s ˆ denotes the invariant component of the discontinuity [Ota88] notation, S 0 ⊂ C domain of the action of ρ0 (π1 (N )). It is a normal planar covering of the exterior boundary ∂e N with deck transformation group ρ0 (π1 (N )). Since ρ0 (π1 (N )) is a ˆ coincides with function group, the limit set Λρ0 of the action of ρ0 (π1 (N )) on C 0 ˆ the boundary of S in C [Mas88].

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4.1. Treelike structure of S 0 We recall that a meridian is a simple closed curve which is nullhomotopic in N but not in ∂e N . By Dehn’s Lemma, a meridian bounds an embedded disk in N . So, every maximal disjoint union of non-parallel meridians cuts N into smaller pieces. On the level of groups, this means that π1 (N ) is a graph of groups whose edge groups are trivial and correspond to the meridians. The universal cover of the graph is a tree and there is a π1 (N )-equivariant map from S 0 to the tree. Such a map maps geodesics in S 0 to paths in the tree. In general, the paths can be fairly arbitrary, in particular not monotone. Definition. Let m be a meridian. An m-wave is an arc on ∂e N with endpoints on m which is homotopic in N relative endpoints, but not in ∂e N to a subarc of m. For example, if two meridians intersect, then each of them contains a wave with respect to the other. Definition. A curve γ : R → ∂e N (resp. γ : R+ → ∂e N ) is in tight position with respect to m if γ does not contain m-waves and the image under γ of every unbounded interval intersects m. Lemma 1. If γ : R+ → ∂e N is a curve which is in tight position with respect to a meridian m, then every lift of γ to S 0 has a well-defined endpoint in the limit set ˆ Λρ0 of the action of ρ0 (π1 (N )) on C. Proof. Let γ 0 : R+ → S 0 be a lift of γ to S 0 . There is a sequence (mi (γ 0 )) of lifts of m which are intersected by γ 0 and indexed by the ordering in γ 0 . Observe ˆ of the that every lift of m to S 0 separates S 0 and denote by Ki the closure in C 0 0 ˆ connected component of C − mi (γ ) that does not contain m0 (γ ). Since γ is in tight position with respect to m, Ki+1 is contained in the interior of Ki for all i ¤ and the diameter of Ki tends to zero (compare [Ota88, 1.9, 1.14]). The following lemma establishes a kind of continuity for the map which associates to lifts of curves γ : R+ → ∂e N which are in tight position with respect to ˆ Notation as in the proof of the a common meridian their endpoints in Λρ0 ⊂ C. last lemma. Lemma 2. For j = 1, . . . , ∞ let γj : R+ → ∂e N be curves which are all in tight position with respect to a meridian m. Let m0 and γj0 be lifts of m and γj to S 0 with m0 = m0 (γj0 ) for all j = 1, . . . , ∞. Then the endpoints of γj0 converge to the 0 if and only if for all i there is j0 such that for all j ≥ j0 endpoint of γ∞ 0 mi (γj0 ) = mi (γ∞ )

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So far, we considered general curves on ∂e N , we now turn our attention to laminations. Definition. A lamination is in tight position with respect to a meridian m if every half-leaf is. Notice that a minimal lamination λ is in tight position with respect to a meridian m if and only if some half-leaf in λ is since every half-leaf is dense. On the other hand, if a lamination consists of several minimal components each being in tight position with respect to a meridian, it is not clear if there is a common meridian with respect to which all components are in tight position. We prove Lemma 3. Let µ be a measured lamination with the property that every component is in tight position with respect to some meridian. Then there is a meridian m intersecting µ and such that µ does not contain any m-wave. Proof. Let µ1 be a component of µ and let m1 be a meridian with respect to which µ1 is in tight position. We claim that the set of those homotopy classes of m1 -waves (rel m1 ) that are represented by a subsegment of a leaf of µ is a finite number Nm1 (µ). For this, remark that there are leaves l2 , . . . , lk which are dense in µ − µ1 . This implies that every m1 -wave in µ can be represented, up to homotopy (rel m1 ), by a segment in one of the leaves l2 , . . . , lk . Recall that each of the leaves l2 , . . . , lk is in tight position with respect to some meridian. It follows from lemma 1 that there are, up to homotopy, only finitely many m1 -waves in l2 , . . . , lk . We have proved that Nm1 (µ) < ∞. We will inductively reduce this number until we obtain a meridian m with Nm (µ) = 0. We assume that Nm1 (µ) > 0; hence, there is an m1 -wave [a, b] contained in one of the leaves l2 , . . . , lk such that (a, b) ∩ m1 = ∅. By surgery of m1 along [a, b] we obtain meridians m2 , m02 with the following properties: (i) µ1 is in tight position with respect to one of the meridians m2 , m02 , say with respect to m2 , and (ii) Nm2 (µ) ≤ Nm1 (µ) − 1. If µ contains m2 -waves, we repeat the process with m2 . After finitely many steps, we are done. ¤ We finish this section with two results of Otal [Ota88]. In the next subsection, we give complete proofs of more precise statements in the case that N is a handlebody. So, we skip the proofs even though Otal’s th`ese d’Etat is unfortunately unpublished. Lemma 4. [Ota88, 1.3] Every minimal component of a lamination λ ∈ O is in tight position with respect to some meridian. ¤ Definition 1. A leaf l : R → ∂e N of a lamination on ∂e N is called homoclinic if

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there are two sequences xi , yi ∈ R and a lift l0 of l to S 0 such that |xi − yi | → ∞ but the distances between l0 (xi ) and l0 (yi ) are bounded in S 0 . Otal proved (see the proof of [Ota88, 2.10]) Lemma 5. A lamination λ in the Masur-domain is not contained in a lamination with a homoclinic leaf. ¤

4.2. The handlebody case This subsection is devoted to a more detailed analysis of laminations on the boundary of the handlebody. As remarked above, the proofs are inspired by the arguments in [Ota88]. Notice that it is a topological property for a lamination to be an element of O. 0 the set of those laminations in PML which have same support as Denote by Mtop laminations in M 0 . The main result of this section is the next Proposition which is a slightly stronger version of Lemma 5 in the handlebody case. Proposition 1. If N is a handlebody and µ ⊂ ∂e N is a lamination containing a 0 . homoclinic leaf, then every minimal component of µ is an element of Mtop On the other hand, every Hausdorff-limit of meridians contains a leaf which is homoclinic (see Casson-Long [CL85] or Otal [Ota88]). Corollary 1. Every minimal component of a Hausdorff-limit of meridians on the 0 . ¤ boundary of a handlebody is an element of Mtop The proof of Proposition 1 occupies the rest of this section. First, we give three 0 . lemmas which help to identify a lamination as an element of Mtop If µ ⊂ ∂e N is a minimal lamination, we denote the smallest compact subsurface of ∂e N containing µ by S(µ). It is unique up to isotopy. Lemma 6. [Ota88, 1.3.2] If µ is a minimal lamination on the boundary of a han0 . dlebody N and ∂e N − S(µ) is compressible, then µ ∈ Mtop Proof. By Dehn’s Lemma [Jac80], ∂e N − S(µ) contains a meridian which bounds a disk in N . If the disk is separating, it cuts N into two handlebodies and µ is contained in one of them, hence there is also a non-separating meridian disjoint from µ. So we assume that m is non-separating. Cut ∂e N along m and join the two resulting boundary components by an embedded arc κ. The boundary of a regular neighbourhood of m ∪ κ in ∂e N is a meridian. Since κ can be chosen as 0 . ¤ close to µ as wanted, we deduce µ ∈ Mtop

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The proof of Lemma 6 fails to generalize to laminations on the boundary of a general compression body and constitutes the main difference between the handlebody case and the general case in the present context. By Lemma 4, every component of a measured lamination in O is in tight position with respect to some meridian. On the other hand we have Lemma 7. Let µ be a minimal arational lamination on the boundary of a handlebody. If µ is not in tight position with respect to any meridian, then 0 . µ ∈ Mtop Proof. Let m be a meridian and l a leaf of µ. As seen before, if l contains only finitely many homotopy classes (rel m) of m-waves, then we can find a meridian m0 with respect to which l is in tight position. The minimality of µ then implies that µ is in tight position with respect to m0 . If l contains infinitely many homotopy classes (rel m) of m-waves, it is homoclinic. So, we find sequences (xi ), (yi ) ⊂ R such that l(xi ), l(yi ) ∈ m and the segments l[xi , yi ] are m-waves. We may suppose that l(xi ) and l(yi ) converge. Fix a transverse measure on µ. For all ² > 0 we can find i, j such that the measure of the small subsegments [l(xi ), l(xj )] ⊂ m and [l(yi ), l(yj )] ⊂ m is less than ². The union of these two segments and the m-waves l[xi , yi ] and l[xj , yj ] is a compressible curve. By the Loop Theorem [Jac80], we find nearby a meridian m² with i(µ, m² ) < 2². Taking limits we obtain a lamination ν ∈ M 0 with i(µ, ν) = 0 which implies that µ and ν have same support as µ is minimal arational. ¤ We now consider the case that µ is only minimal. If some component of ∂S(µ) 0 . If this is not the case, a relative is a meridian, Lemma 6 shows that µ ∈ Mtop version of the proof of Lemma 7 yields the following Lemma which is essentially a special case of Theorem 1.6 in Otal [Ota88]. Lemma 8. If a minimal lamination µ on the boundary of a handlebody is not in tight position with respect to any meridian, then S(µ) is compressible and µ ∈ 0 . ¤ Mtop We deduce from Lemma 8 and Lemma 6 Corollary 2. If a minimal component of a lamination µ on the boundary of a handlebody is not in tight position with respect to any meridian, then every minimal 0 . ¤ component of µ is in Mtop

Proof of Proposition 1. A similar argument as in the proof of Lemma 7 shows that every ²-neighbourhood of a homoclinic leaf l : R → ∂e N in µ contains a meridian. Lemma 6 implies that every minimal component disjoint from the closure of l is 0 . in Mtop

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From now on, we assume that l is dense in µ. So µ has one or two minimal components. If one of them is not in tight position with respect to some meridian, we are done by Corollary 2. Assume that they are in tight position with respect to some meridian. Since a homoclinic leaf is not in tight position with respect to any meridian, the leaf l must be isolated and non-closed. Further, every lift l0 of l ˆ which coincide since l is homoclinic. to S 0 has endpoints in the limit set Λρ0 ⊂ C If µ contains two minimal components, then every meridian with respect to which one is in tight position intersects the other. Thus, Lemma 3 provides a meridian m with respect to which every minimal component of µ is in tight position. In particular, we find x0 < y0 ∈ R such that the segment l[x0 , y0 ] is an m-wave and such that the half-leaves l|{t≤x0 } and l|{t≥y0 } are in tight position with respect to m. Let x0 > x1 > x2 > . . . and y0 < y1 < y2 < . . . be the sequences of all points with l(xi ) ∈ l ∩ m and l(yi ) ∈ l ∩ m. We have for all i < j: (1) The segment l[xi , yi ] is an m-wave, d d (2) the tangent vectors dt l|yi and − dt l|xi point to the same side of m, and (3) the curves l[xi , xj ] and l[yi , yj ] represent the same element in π1 (N, D) where D is a disk with ∂D = m. We will first treat the case that µ has only one minimal component µ0 . For every train track τ carrying µ = µ0 ∪ l we will construct a simple closed curve in M 0 carried by τ . As l is not contained in the support of any measured lamination this 0 . will prove that µ0 ∈ Mtop Suppose first that d(l(xi ), l(yi )) ≥ ² > 0 for all i. After refining τ , we may assume that every component of τ ∩ m is shorter than ². If two points a, b are contained in the same component of τ ∩ m we denote by [a, b] the subsegment of this component bounded by a and b. There are i < j such that one of the following two cases occurs. Case I: (1) Each of the pairs {l(xi ), l(xj )} and {l(yi ), l(yj )} is contained in a component of τ ∩ m, d d (2) the tangent vectors − dt l|xi and − dt l|xj point to the same side of m, and (3) there is no k ∈ {i + 1, . . . , j − 1} such that l(xk ) or l(yk ) belongs to [l(xi ), l(xj )]. Case II: (1) Each of the pairs {l(xi ), l(yj )} and {l(yi ), l(xj )} is contained in a component of τ ∩ m, d d l|xi and dt l|yj point to the same side of m, and (2) the tangent vectors − dt (3) there is no k ∈ {i + 1, . . . , j − 1} such that l(xk ) or l(yk ) belongs to [l(xi ), l(yj )]. Both cases are represented in Figure 1. If Case I holds, there is no k such that l(xk ) and l(yk ) belong to [l(yi ), l(yj )] because d(l(xk ), l(yk )) ≥ ². Together with the fact that lifts of l to S 0 do not

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ˆ this implies that there is intersect and can be compactified to Jordan curves in C no k ∈ {i + 1, . . . , j − 1} such that l(xk ) or l(yk ) lies in [l(yi ), l(yj )]. So, the curves γa = l[xi , xj ] ∪ [l(xi ), l(xj )], ηa = l[yi , yj ] ∪ [l(yi ), l(yj )] are homotopic to simple closed geodesics carried by τ . They belong to M 0 since a boundary component of a regular neighbourhood of γa ∪ ηa ∪ l[xi , yi ] is a meridian disjoint from γa and ηa . In Case II, we deduce as in Case I that there is no k ∈ {i + 1, . . . , j − 1} such that l(xk ) or l(yk ) belongs to [l(yi ), l(xj )]. As above this implies that the curve γb = l[xi , xj ] ∪ [l(xj ), l(yi )] ∪ l[yi , yj ] ∪ [l(yj ), l(xi )] is homotopic to a simple closed geodesic carried by τ and belongs to M 0 because it is disjoint from the boundary component of a neighbourhood of γb ∪ l[xi , yi ] which is a meridian. The arguments are valid for any train track τ . We conclude that the minimal 0 if d(l(xi ), l(yi )) ≥ ² > 0 for all i. component of µ is in Mtop Continuing with the assumption that µ contains only one minimal component µ0 , suppose that inf d(l(xi ), l(yi )) = 0. In particular, l contains m-waves whose endpoints are close and we cannot directly apply the same arguments as before. Fix i such that l(xi ), l(yi ) are contained in the same component of τ ∩ m. If there is no j > i with l(xj ), l(yj ) ∈ [l(xi ), l(yi )] we finish the proof by Lemma 6 as l[xi , yi ] ∪ [l(xi ), l(yi )] is compressible and disjoint from µ0 . Otherwise, let j > i be minimal for the property that l(xj ) and l(yj ) belong to [l(xi ), l(yi )]. Figure 2 represents the four possible cases. By the choice of i, j and the fact that lifts of l to S 0 do not intersect and can ˆ the Cases I and II can be treated as above. be compactified to Jordan curves in C, The same argument shows in Case III that a boundary component of a regular neighbourhood of l[xi , xj ] ∪ [l(xj ), l(xi )] ∪ l[xi , yi ] ∪ l[yi , yj ] ∪ [l(yj ), l(yi )]

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is homotopic to a simple closed geodesic carried by τ . It is even a meridian since the segments l[xi , xj ] and l[yi , yj ] represent the same element in π1 (N, D). Similary, in Case IV, the curve γd which is the concatenation of the segments l[xi , xj ], [l(xj ), l(yi )], l[yi , xi ], [l(xi ), l(yj )], l[yj , yi ] and l[yi , xi ] is homotopic to a meridian carried by τ . As before, we conclude that the minimal component of µ 0 . is in Mtop The remaining case that µ has two minimal components can be reduced to the discussion of Case I because there is obviously some ² > 0 such that d(l(xi ), l(yj )) ≥ ² > 0 for all i, j. Notice that each of the curves γa and ηa constructed above approximates one of the minimal components of µ and is in M 0 . This implies that 0 . ¤ both minimal components of µ are in Mtop

5. Morphisms This chapter is the central part of the proofs of Theorem 3 and Corollary 3. A large part is inspired by ideas of Skora [Sko96]. N is again a compression body and ρ0 is a convex cocompact representation of π1 (N ) which uniformizes N . Definition. Let T 0 , T be R-trees. A morphism from T 0 to T is a map Φ : T 0 −→ T with the property that every non-degenerate arc [p, q] ⊂ T 0 contains a nondegenerate subarc [p, r] ⊂ [p, q] which is mapped isometrically onto Φ[p, r] ⊂ T . A morphism is said to fold at a point p ∈ T and p is a folding point, if there are non-degenerate arcs [p, q], [p, q 0 ] ⊂ T 0 , [p, q] ∩ [p, q 0 ] = {p} with Φ[p, q] = Φ[p, q 0 ].

5.1. Morphisms from dual trees In section 2, we discussed trees which are dual to measured laminations. If µ ∈ ML, the dual tree Tµ can be seen as the leaf space of a measured partial foliation Fµ enlarging µ and we denote the projection by πFµ : H2 → Tµ . A morphism from a dual tree Tµ to another tree T is said to fold only at complementary regions if the only folding points are projections of complementary regions of Feµ ⊂ H2 . The existence of such morphisms is fundamental in the proof of Skora’s Theorem. The following theorem is essentially a special case of the main result in [MO93] (see also [Ota96, chapter 8]). Theorem (Morgan–Otal). Let (α1 , . . . , α3g−3 ) be a collection of simple closed curves which define a pants decomposition of a closed surface S and let π1 (S) y T be an action on an R-tree T . Then there is a measured lamination µ ∈ ML and

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an equivariant morphism Φ : Tµ −→ T with lT (αi ) = lTµ (αi ) for all i. Moreover, Φ folds only at complementary regions. Remark. Notice that if lT (αi ) > 0, then Φ maps the axis of αi in Tµ isometrically to the axis of αi in T [Kap00]. The second step in the proof of Skora’s Theorem is to study equivariant morphisms from dual trees Tµ , µ ∈ ML, to a tree with a minimal small action of the fundamental group of a closed surface which fold only at complementary regions. Skora proves that such a morphism is an isometry, and in particular µ is unique. In the present situation, the actions we will consider are not even effective because they factor through the compression homomorphism ϕ : π1 (∂e N ) → π1 (N ). We prove Proposition 2. Let N be a compression body with exterior boundary surface ∂e N . Let π1 (N ) y T be a minimal small action and let µ be a measured lamination on ∂e N such that there is an equivariant morphism Φ : Tµ −→ T that folds only at complementary regions. Then µ is not in the Masur domain O. Moreover, if N is a handlebody, then every minimal component of µ is an 0 . element of Mtop Proof. If some component of µ is not in tight position with respect to any meridian, then µ ∈ / O by Lemma 4. Moreover, if N is a handlebody we are done by Corollary 2. We assume from now on that every component of µ is in tight position with respect to some meridian. Then by Lemma 3, there is a meridian m intersecting µ such that µ does not contain any m-wave. By definition, every component of µ intersecting m is in tight position with respect to m. We will show that we can extend µ to a lamination with a homoclinic leaf. In this case, Lemma 5 shows that µ ∈ / O. Moreover, Proposition 1 proves that every 0 if N is a handlebody. component of µ is in Mtop e Lemma 9. Φ folds along m: for a lift m e of m to H2 there are segments I˜1 , I˜2 ⊂ m intersecting in a single point ˜ ? such that Φ maps πFµ (I˜1 ) and πFµ (I˜2 ) isometrically onto a non-degenerate segment J in T . e is invariant under Proof. Let x be a point on a lift m e of m to H2 . The geodesic m some m ∈ π1 (∂e N ) which is trivial in π1 (N ). By equivariance Φ(πFµ (mx)) = Φ(πFµ (x))

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The morphism Φ folds the non-degenerate segment [πFµ (x), πFµ (mx)] only finitely many times. So we can find small segments I˜1 , I˜2 ⊂ (x, mx) such as in the statement. ¤ Remark. Note that the claim is true for every meridian which intersects µ. We continue with the proof of Proposition 2 and choose a transverse orientation on m. For a point x ∈ m ∩ µ we use the symbol µ+ x for the half-leaf of µ starting is in tight position with respect to at x with positive direction. By definition, µ+ x m. The following Proposition will be proved at the end of this section. Proposition 3. For every pair of intervals (I1 , I2 ) as in Lemma 9, there are zi ∈ + Ii ∩ µ (i = 1, 2) such that the lifts to S 0 of µ+ z1 and µz2 have the same endpoints ˆ in the limit set Λρ0 of ρ0 (π1 (N )) in C. We fix a pair of intervals (I˜10 , I˜20 ) as in Lemma 9. As Φ folds only at complementary regions, the point ˜ ? lies in a complementary region C? of Feµ . After the ˜, too. There collapse of Fµ to µ, C? can be seen as a complementary region of µ 0 ˜ ˜2 of C? with µ ˜i ∩ Ii 6= ∅. They are are two well determined boundary leaves µ ˜1 , µ different because separated by the complementary region C? . Up to reversing the ˜+ orientation, we may assume that µ ˜+ 1 and µ 2 are not asymptotic. Next we show that the endpoints in Λρ0 of their projections to S 0 are equal. We choose a sequence of nested intervals (I˜1k , I˜2k )k∈N as in Lemma 9 such that i(I˜1k , Feµ ) = i(I˜2k , Feµ ) > 0 tends to zero. By Proposition 3, for every k ∈ N and and µ+ have the same i = 1, 2 we get zik ∈ Iik ∩ µ such that the lifts to S 0 of µ+ z1k z2k endpoint in the limit set Λρ0 . Since for i = 1, 2 and k −→ ∞ the sequence i(I˜ik , Feµ ) tends to zero, the ) tends to µ+ sequence (µ+ i , the projection to ∂e N of the boundary half-leaf z k k∈N i

+ + 0 µ ˜+ i . By Lemma 2, the lifts of µ1 and µ2 to S have the same endpoint in Λρ0 . ˜+ ˜+ Now let l be the geodesic in H2 joining the endpoints in ∂H2 of µ 1 and µ 2 . The ˜ ∪l is a geodesic lamination µl and l projects to a homoclinic projection to ∂e N of µ leaf, since the endpoints of the projection to S 0 coincide. So, µ is contained in a lamination with a homoclinic leaf. This finishes the proof of Proposition 2. ¤

The rest of this section is devoted to the proof of Proposition 3.

5.2. Skora’s argument We use the same notation as above, Φ : Tµ → T is the morphism of Proposition 2, Fµ is an enlargement of µ, m is a meridian with i(m, µ) > 0 and such that every

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component of µ intersected by m is in tight position with respect to it and I˜1 , I˜2 , ˜? are as in Lemma 9. Denote the projections to the surface ∂e N by I1 , I2 , ?. Recall that ˜ ? is in a complementary region C? of Feµ ⊂ H2 . Fix a transverse orientation on m. Take the oriented first return maps of Fµ [Ota96, Sko96] Ai : Ii ∩ Fµ → Ii ∩ Fµ for i = 1, 2. Ai induces an interval exchange transformation on πFµ (I˜i ) ⊂ Tµ . Conjugating A1 , A2 by Φ one obtains two interval exchange transformations B1 , B2 on J = (Φ ◦ πFµ )(I˜1 ) = (Φ ◦ πFµ )(I˜2 ) ⊂ T [Sko96]. Let C denote the free semi-group generated by two letters c1 , c2 and B the semi-group generated by the maps B1 , B2 . There is an obvious homomorphism θ : C −→ B, c = c²1 · . . . · c²n 7−→ θ(c) = B²n ◦ . . . ◦ B²1 . The set of boundary leaves of Feµ is countable [Ota96]. A point z ∈ T such that none of its preimages under Φ is represented by a boundary leaf of Feµ is called regular. If z ∈ J is a regular point, we denote by z˜i for i = 1, 2 the point I˜i ∩ (Φ ◦ πFµ )−1 (z). The projection of z˜i to ∂e N is denoted zi . A regular point z ∈ J and a letter c² ∈ {c1 , c2 } determine a closed curve γz,² ⊂ ∂e N which is the concatenation of (1) the subsegment of I² from ? to z² , (2) the subsegment of the half-leaf of Fµ from z² to A² z² in positive direction, and (3) the subsegment of I² from A² z² to ?. Let z ∈ J be regular such that for all B ∈ B, Bz is regular, too. For c = c²1 · . . . · c²n ∈ C we define the closed curve ωz (c) in ∂e N to be the concatenation of the curves γ²1 ,z , γ²2 ,B²1 z , . . . , γ²n ,B²n−1 ◦...◦B²1 (z) , that is ωz (c) = γ²1 ,z ∗ γ²2 ,B²1 z ∗ . . . ∗ γ²n ,B²n−1 ◦...◦B²1 (z) .

(3)

Recall that Fµ is obtained from µ by blowing up closed leaves of µ to fibered collars. After collapsing the collars back to closed leaves of µ, we denote the image of the curve ωz (c) by ωz (c) as well. After a small homotopy near m this curve and m can be made transverse. Moreover, it is in tight position with respect to m. The element in π1 (∂e N, ?) represented by ωz (c) will be denoted [ωz (c)]. The map [ωz (·)] : C → π1 (∂e N, ?) is injective because ˜? is in a complementary region [Sko96]. It is not a homomorphism but we have the following equation, which follows from the definition: [ωz (a · b)] = [ωz (a)] ∗ [ωθ(a)z (b)] for all a, b ∈ C.

(4)

The action of ϕ([ωz (c)]) ∈ π1 (N ) on the tree T is related to the interval exchange map θ(c) by (see [Sko96]) θ(c) = ϕ([ωz (c)]) on a small neighbourhood of z in J.

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Lemma 10. For every pair of intervals (I1 , I2 ) as in Lemma 9 and every k there is a regular point z k ∈ J and ak1 = ck1 · . . . , ak2 = ck2 · . . . ∈ C such that the curves ωzk (ak1 ), ωzk (ak2 ) represent the same element in π1 (N, ?). The proof of this Lemma is essentially the proof of Proposition 3.1 in [Sko96]. We refer to it for a more detailed exposition. Proof. Let z ∈ J be regular such that for all B ∈ B, Bz is regular, too. We fix k ∈ N. Let Cnk ⊂ C denote the words of length n in the letters ck1 = c1 · . . . · c1 and ck2 = c2 · . . . · c2 . Cardinality of Cnk is 2n . As in [Sko96], the cardinality of the sets θ(Cnk )(z) only grows polynomially in n. So, for fixed k there is a sequence of sets (Dnk )n , with Dnk ⊂ Cnk such that: (1) for fixed n any two elements of θ(Dnk ) map z to the same point in J and (2) the cardinality of Dnk has exponential growth in n. We fix dkn ∈ Dnk . For fixed n, k the elements of the set (θ(dkn ))−1 ◦θ(Dnk ) are interval exchange maps on J which fix a common segment around z. By smallness of the action π1 (N ) y T and equation (5), the set ϕ([ωz (Dnk )] ∗ [ωz (dkn )]−1 ) is contained in a cyclic subgroup Znk ⊂ π1 (N ) [Sko96]. The subgroups can be chosen to satisfy k ⊂ ... Znk ⊂ Zn+1 but such a sequence stabilizes and we find a cyclic subgroup Z k ⊂ π1 (N ) such that for all n ϕ([ωz (Dnk )] ∗ [ωz (dkn )]−1 ) ⊂ Z k As in [Sko96], the cardinality of ϕ([ωz (Dnk )]∗[ωz (dkn )]−1 ) has at most linear growth in n. So there is n depending on k and there are different words bk1 , bk2 ∈ Dnk ⊂ Cnk with ϕ([ωz (bk1 )]) = ϕ([ωz (bk2 )]) The words bk1 , bk2 ∈ Cnk are different but may coincide at the beginning. Without loss of generality we can assume that bk1 = αk · ck1 · . . . = αk · ak1 bk2 = αk · ck2 · . . . = αk · ak2 with αk being a word in ck1 , ck2 of length less than n. By equation (4) [ωz (bk1 )] = [ωz (αk )] ∗ [ωθ(αk )z (ak1 )] [ωz (bk2 )] = [ωz (αk )] ∗ [ωθ(αk )z (ak2 )] We set z k = θ(αk )z and we are done.

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Recall that we have fixed a transverse orientation on m and that for a point x ∈ m ∩ µ the half-leaf µ+ x of µ starting at x with positive direction is in tight position with respect to m. Proof of Proposition 3. Notation as in Lemma 10. For i = 1, 2 let z˜ik ∈ I˜i ∩ Feµ be the unique point which is mapped to z k ∈ J by Φ ◦ πFµ and zik the projection to ∂e N . After collapsing the collars of Fµ back to closed leaves of µ we denote the images of zik by zik again (compare above). In the proof of Lemma 1, we defined the sequence (mj ) of lifts of m to S 0 for curves which are in tight position with respect to a common meridian. The curves ωzk (ak1 ) and ωzk (ak2 ) provided by Lemma 10 are in tight position with respect to the meridian m. Since they represent the same element in π1 (N, ?) mj (ωzk (ak1 )) = mj (ωzk (ak2 )) for all j, k

(6)

By construction of the curves ωzk (aki ), we have (i = 1, 2) ) for all k and j = 1, . . . , k mj (ωzk (aki )) = mj (µ+ zk i

(7)

Equations (6) and (7) imply ) = mj (µ+ ) for all k and j = 1, . . . , k mj (µ+ zk zk 1

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By compactness of Ii ∩ µ, we can suppose that the sequence (zik )k converges to + 0 some zi ∈ Ii ∩ µ. By Lemma 2, the lifts of µ+ z1 and µz2 to S have the same ¤ endpoint in the limit set Λρ0 .

6. Realizations Let N be a compression body with a convex cocompact hyperbolic structure. Recall that every lamination λ in O is realized by a pleated surface with respect to ρ0 and that the induced map from λ to the projectivized tangent bundle of N is a homeomorphism onto its image Pλ . By definition, the lamination λ ∈ O is realized in a tree T if there is a continuous eλ , the lift of Pλ to the projectivized tangent bundle and equivariant map from P 3 of H , to the tree T which is injective when restricted to any leaf. Since λ is mapped homeomorphically onto Pλ it suffices to find a continuous ˜ the lift of λ to H2 , to the tree T which is and π1 (∂e N )-equivariant map from λ, injective when restricted to any leaf. By abuse, if such a map exists we will also say that λ is realized in T . Theorem 3. Let π1 (N ) y T be a non-trivial minimal small action on an R-tree T and λ a minimal arational measured lamination in the Masur domain, then λ is realized in T .

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The idea of the proof is to show that there is a measured lamination µ on the exterior boundary ∂e N and a morphism Φµ : Tµ → T such that the composition of the projection H2 → Tµ with Φµ can be deformed to a realization of λ in T . Proof. We choose a sequence of simple closed curves (γi )i converging to λ with respect to the Hausdorff topology. By Morgan’s and Otal’s Theorem, for all i, there is a measured lamination µi with i(µi , γi ) = lTµi (γi ) = lT (γi ) and a morphism Φµi : Tµi → T. Now let us show that lT (γi ) > 0 for large i. Up to choice of a subsequence, we may assume that the supports of the laminations µi converge in the Hausdorff topology to a lamination µH . By Proposition 2, the measured laminations µi are not in the Masur domain O. This implies that the Hausdorff limit µH cannot contain a lamination in O. In particular, the minimal arational lamination λ intersects µH . So, for all but finitely many i, say for all, one has lT (γi ) = i(µi , γi ) > 0.

(8)

Remark that the morphism Φµi maps isometrically the axis of every element in the conjugacy class represented by γi . For all i, we choose an enlargement Fµi of µi in order to obtain a continuous and equivariant projection πFµi : H2 → Tµi . We may assume that the enlargements Fµi converge in the Hausdorff topology to the lamination µH . The map Φµi ◦ πFµi is continuous, equivariant and, by equation (8), it is monotone on every lift of γi to H2 . Here, a map from an interval to a tree is monotone if the preimage of every point is at most a bounded interval [Ota96]. Lemma 11. There is a train track τ which carries λ such that, for large i, the map Φµi ◦ πFµi satisfies: (i) It is constant on ties, and monotone and non-constant on the rails of every ej ⊂ H2 of every rectangle Rj ⊂ τ . lift R ej , R ek ⊂ H2 which meet in vertical sides (ii) The images of any two rectangles R intersect in exactly one point. Proof. Since λ intersects µH , the construction in [Ota96, chapter 3] yields a train track τ carrying λ such that, for large i, the projection πFµi : H2 → Tµi is constant ej ⊂ H2 of every on ties, and monotone and non-constant on the rails of every lift R rectangle Rj ⊂ τ . ej , R ek ⊂ We may assume that τ carries λ minimally, i.e. for any two rectangles R 2 ˜ e e H which meet in vertical sides one has Rj ∩ Rk ∩ λ 6= ∅. For large i, the train track τ carries the curve γi minimally, too. We are going to show that for all such i, the map Φµi ◦ πFµi satisfies (i) and (ii). ej ; hence, we can ej , there is a lift γ˜i ⊂ H2 of γi crossing R Given a rectangle R e homotope along ties the rails of Rj into the lift γ˜i . As remarked above, the map

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ej . It is not constant Φµi ◦ πFµi is monotone on γ˜i , and thus on the rails of R ej ) is a non-degenerate segment and Φµ is a morphism. Property because πFµi (R i (i) follows. ek ⊂ H2 which meet in vertical sides there is a lift ej , R Given two rectangles R 2 e ek 6= ∅. Again, since Φµ ◦ πF is monotone on γ˜i , γ˜i ⊂ H of γi with γ˜i ∩ Rj ∩ R i µi property (ii) follows. ¤ For τ and i as in lemma 11, it follows from [Ota96, 3.1.5, 3.1.6] that the map Φµi ◦ πFµi can be homotoped to a realization in T of every lamination carried by τ , in particular of λ. This concludes the proof of Theorem 3. ¤ In the particular case that N is a handlebody we obtain Theorem 4. Let N be a handlebody and π1 (N ) y T a non-trivial minimal small action on an R-tree T . Further let λ0 be a minimal component of a lamination λ in the Masur domain. Either λ0 is realized in T or there is a train track τ0 carrying λ0 and a continuous and equivariant map Φ0 : τ˜0 → T that maps every connected component of τ˜0 to a point. Proof. If λ0 is a simple closed geodesic, then there is nothing to prove. So suppose it is not. As above choose a sequence of simple closed curves (γi )i converging to λ0 with respect to the Hausdorff topology. Again, for all i, there is a measured lamination µi with i(µi , γi ) = lTµi (γi ) = lT (γi ) and a morphism Φµi : Tµi → T. Suppose that the laminations µi converge in the Hausdorff topology to a lamination µH . Every minimal component of µH is contained in the Hausdorff limit of components of µi , thus by Proposition 2 in a Hausdorff limit of laminations 0 and so in a Hausdorff limit of meridians. By Corollary 1, every minimal in Mtop 0 . component of µH is an element of Mtop So either λ0 is transverse to µH or disjoint from µH . In the first case we conclude as in the proof of Theorem 3. If λ0 ∩ µH = ∅, choose a train track τ0 carrying λ0 and disjoint from µH . For i large enough, there is a partial foliation Fµi enlarging the measured lamination µi such that τ0 is also disjoint from Fµi . This implies that the image of every connected component of τ˜0 is mapped by ¤ πFµi to a point in Tµi . Define Φ0 to be Φµi ◦ πFµi .

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Corollary 3. Let N be a handlebody and π1 (N ) y T be a non-trivial minimal small action on an R-tree T . At least one minimal component of every measured lamination in the Masur domain is realized in T . Proof. Suppose the claim is false. By Theorem 4, every minimal component λj (j = 1, . . . , k) of λ is carried by a train track τj and there is a continuous and equivariant map Φj : τ˜j → T that maps every connected component of τ˜j to a single point. After refining the train tracks, we may assume that they are pairwise disjoint. Since O is open there are simple closed curves ηj carried by τj such that the multicurve η = η1 ∪ · · · ∪ ηk is in the Masur domain. By Morgan’s and Otal’s Theorem, we find a measured lamination µ and a morphism Φµ : Tµ → T such that 0 = lT (ηj ) = lTµ (ηj ) = i(µ, ηj ) for all j = 1, . . . , k. This contradicts the fact 0 whence that by Proposition 2 every minimal component of µ is an element of Mtop i(µ, η) > 0. ¤

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Gero Kleineidam Universit¨ at Bonn Mathematisches Institut Beringstr. 1 D–53115 Bonn Germany e-mail: [email protected]

Juan Souto Universit¨ at Bonn Mathematisches Institut Beringstr. 1 D–53115 Bonn Germany e-mail: [email protected]

(Received: November 30, 2000)

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